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Exponential increase problem

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Exponential increase problem

by jcm91186 » Mon Jun 04, 2012 1:49 pm
I'm looking for a quick way to solve a problem like in the following.

In 1980 John's salary was $15,000 a year and Don's salary was $20,000 a year. If every year thereafter, John received a raise of $2,450 and Don receives a raise of $2,000, the first year in which John's salary will be more than Don's salary is
A - 1987
B - 1988
C - 1991
D - 1992
E - 2000

The way I used in the practice test took far too long to get the answer! Please help me find a quick solution for a problem like this.
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by Mike@Magoosh » Mon Jun 04, 2012 4:16 pm
jcm91186 wrote:I'm looking for a quick way to solve a problem like in the following.

In 1980 John's salary was $15,000 a year and Don's salary was $20,000 a year. If every year thereafter, John received a raise of $2,450 and Don receives a raise of $2,000, the first year in which John's salary will be more than Don's salary is
A - 1987
B - 1988
C - 1991
D - 1992
E - 2000

The way I used in the practice test took far too long to get the answer! Please help me find a quick solution for a problem like this.
I'm happy to help with this. :)

First of all, notice -- this is not an "exponential increase" -- an exponential increase occurs when you are multiplying the same factor each time, but here you are simply adding the same thing each time.

So the quick way to dispatch this problem...

John makes $15K, and Don, $20K, so John has to recoup a $5000 deficit. Note that and put it aside a moment.

Each year, John gets a $2450 raise, and Don, a $2000, so every year, John gains $2450 - $2000 = $450 on Don.

John is $5000 behind Don and is gaining at a rate of $450 a year. After 10 years, he has made up $4500 of the difference --- still $1000 behind Don. After 11 years, he makes up $4500 + $450 = $4950, so he's still $50 short of Don. After 12 years, he makes up $4950 + $450 = $5400, which is more than $5000, so that's the first time he passes Don --- by $400. The twelfth year is the first year in which John's salary exceeds Don's. That's 1992, answer = D.

Does that approach make sense? For free, here's a lesson video you may find helpful
https://gmat.magoosh.com/lessons/265-int ... d-problems

Let me know if you have any further questions.

Mike :)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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by dhonu121 » Thu Jun 07, 2012 8:00 pm
Easiest way.
Let's assume that after n years their salary become equal.
then
15,000+2450n = 20,000+2000n
on solving
450n = 5000
n = 11.11 years.
Hence rounding off to 12 years.
Answer:1980+12 = 1992
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